If 2x + 3y = 14 and 2x − 3y = 2, find the value of xy.


If 2x + 3y = 14 and 2x − 3y = 2, find the value of xy.

[Hint: Use (2x + 3y)2 − (2x − 3y)2 = 24xy]


We will use the identity $(a+b)(a-b)=a^{2}-b^{2}$ to obtain the value of $x y$.

Squaring $(2 x+3 y)$ and $(2 x-3 y)$ both and then subtracting them, we get:

$(2 x+3 y)^{2}-(2 x-3 y)^{2}=\{(2 x+3 y)+(2 x-3 y)\}\{(2 x+3 y)-(2 x-3 y)\}=4 x \times 6 y=24 x y$

$\Rightarrow(2 x+3 y)^{2}-(2 x-3 y)^{2}=24 x y$

$\Rightarrow 24 x y=(2 x+3 y)^{2}-(2 x-3 y)^{2}$

$\Rightarrow 24 x y=(14)^{2}-(2)^{2}$

$\Rightarrow 24 x y=(14+2)(14-2) \quad\left(\because(a+b)(a-b)=a^{2}-b^{2}\right)$

$\Rightarrow 24 x y=16 \times 12$

$\Rightarrow x y=\frac{16 \times 12}{24}$               (Dividing both sides by 24 )

$\Rightarrow x y=8$

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